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<h1 id="Mesh-Smoothing-and-Optimization">Mesh Smoothing and Optimization<a class="anchor-link" href="#Mesh-Smoothing-and-Optimization">&#182;</a></h1>
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<h2 id="Improve-geometric-mesh-quality">Improve geometric mesh quality<a class="anchor-link" href="#Improve-geometric-mesh-quality">&#182;</a></h2><p>The function</p>

<pre><code>[node,elem] = optmesh(node,elem)

</code></pre>
<p>will optimize the shape
regularity of triangles in the input mesh <code>(node,elem)</code> and outputs a
better mesh <code>(node,elem)</code>. Note that the connectivity, i.e., <code>elem</code> might be changed. Use</p>

<pre><code>node = meshsmoothing(node,elem)

</code></pre>
<p>to move nodes only.</p>
<p>After showing an example, we shall explain algorithms implemented in <code>optmesh.m</code>.</p>

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<div class=" highlight hl-matlab"><pre><span></span><span class="n">load</span> <span class="n">airfoilperturbmesh</span>
<span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span> <span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> 
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">title</span><span class="p">(</span><span class="s">&#39;original mesh&#39;</span><span class="p">);</span>
<span class="n">figure</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span> <span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> 
<span class="n">showmeshquality</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">axis</span><span class="p">([</span><span class="mi">0</span> <span class="mi">1</span> <span class="mi">0</span> <span class="mi">2700</span><span class="p">]);</span>
<span class="p">[</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">]</span> <span class="p">=</span> <span class="n">optmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">);</span> <span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> 
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">title</span><span class="p">(</span><span class="s">&#39;smoothed mesh&#39;</span><span class="p">);</span>
<span class="n">figure</span><span class="p">(</span><span class="mi">2</span><span class="p">);</span> <span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> 
<span class="n">showmeshquality</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">axis</span><span class="p">([</span><span class="mi">0</span> <span class="mi">1</span> <span class="mi">0</span> <span class="mi">2700</span><span class="p">]);</span>
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<pre> - Min quality 0.2456 - Mean quality 0.8953 
Mesh quality before optimization 
 - Min quality 0.2456 - Mean quality 0.8953 
Mesh quality after optimization 
 - Min quality 0.5756 - Mean quality 0.9356 
 - Min quality 0.5756 - Mean quality 0.9356 
</pre>
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<h2 id="ODT-based-mesh-smoothing">ODT-based mesh smoothing<a class="anchor-link" href="#ODT-based-mesh-smoothing">&#182;</a></h2><p>In the function <code>meshsmoothing</code>, we move one node at a time inside its
patch, which consists of all triangles surrounding this node, such that
the interpolation error to the quadratic function $z=x^2+y^2$ is minimized. The function <code>meshsmoothing</code> will keep the topology of the input mesh, i.e., the node index and connectivity of nodes are unchanged.</p>
<p>In the simplest case, the scheme is to move the node to the average of
circumenters of triangles in the local patch. Details can be found in my research webpage <a href="http://math.uci.edu/~chenlong/mesh.html">ODT mesh</a>.</p>

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<div class=" highlight hl-matlab"><pre><span></span><span class="n">theta</span> <span class="p">=</span> <span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="nb">pi</span><span class="o">/</span><span class="mi">3</span> <span class="o">-</span><span class="nb">pi</span><span class="o">/</span><span class="mi">3</span> <span class="mi">0</span> <span class="nb">pi</span><span class="o">/</span><span class="mi">3</span> <span class="mi">2</span><span class="o">*</span><span class="nb">pi</span><span class="o">/</span><span class="mi">3</span> <span class="nb">pi</span><span class="p">]</span><span class="o">&#39;</span><span class="p">;</span>
<span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="nb">cos</span><span class="p">(</span><span class="n">theta</span><span class="p">),</span> <span class="nb">sin</span><span class="p">(</span><span class="n">theta</span><span class="p">)];</span>
<span class="n">node</span><span class="p">(</span><span class="k">end</span><span class="o">+</span><span class="mi">1</span><span class="p">,:)</span> <span class="p">=</span> <span class="mi">0</span><span class="p">;</span>
<span class="n">elem</span> <span class="p">=</span> <span class="n">delaunayn</span><span class="p">(</span><span class="n">node</span><span class="p">);</span>
<span class="n">node</span><span class="p">(</span><span class="k">end</span><span class="p">,:)</span> <span class="p">=</span> <span class="nb">rand</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span><span class="o">*</span><span class="mf">0.4</span><span class="p">;</span>
<span class="n">figure</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">);</span>
<span class="n">c</span> <span class="p">=</span> <span class="n">circumcenter</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">hold</span> <span class="n">on</span><span class="p">;</span> <span class="n">plot</span><span class="p">(</span><span class="n">c</span><span class="p">(:,</span><span class="mi">1</span><span class="p">),</span><span class="n">c</span><span class="p">(:,</span><span class="mi">2</span><span class="p">),</span><span class="s">&#39;r.&#39;</span><span class="p">,</span><span class="s">&#39;MarkerSize&#39;</span><span class="p">,</span><span class="mi">16</span><span class="p">)</span>
<span class="n">node</span><span class="p">(</span><span class="k">end</span><span class="p">,:)</span> <span class="p">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">c</span><span class="p">);</span>
<span class="n">plot</span><span class="p">(</span><span class="n">node</span><span class="p">(</span><span class="k">end</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="n">node</span><span class="p">(</span><span class="k">end</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="s">&#39;b.&#39;</span><span class="p">,</span><span class="s">&#39;MarkerSize&#39;</span><span class="p">,</span><span class="mi">16</span><span class="p">)</span>
<span class="n">figure</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span>
<span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span> <span class="n">findnode</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="s">&#39;all&#39;</span><span class="p">,</span><span class="s">&#39;noindex&#39;</span><span class="p">);</span>
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<h2 id="Edge-swapping">Edge swapping<a class="anchor-link" href="#Edge-swapping">&#182;</a></h2>
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<p>Edge swapping is used to further improve the mesh quality. <code>elem = edgeswap(node,elem)</code> swap diagonals of a convex quadlateral formed by two triangles of the mesh (node,elem) such that the summation
of opposite angles are less than or equal to 180 degree, i.e., it is locally Delaunay.</p>
<p>Since the connectivity may be changed, some elementwise quantity, e.g.,
the material property, if any, should be updated accordingly.</p>

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<div class="prompt input_prompt">In&nbsp;[4]:</div>
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<div class=" highlight hl-matlab"><pre><span></span><span class="n">node</span> <span class="p">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">;</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">;</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">];</span>
<span class="n">elem</span> <span class="p">=</span> <span class="p">[</span><span class="mi">1</span> <span class="mi">2</span> <span class="mi">4</span><span class="p">;</span> <span class="mi">4</span> <span class="mi">3</span> <span class="mi">1</span><span class="p">];</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span> <span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">elem</span> <span class="p">=</span> <span class="n">edgeswap</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
<span class="n">subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">);</span> <span class="n">showmesh</span><span class="p">(</span><span class="n">node</span><span class="p">,</span><span class="n">elem</span><span class="p">);</span>
</pre></div>

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